Diagonal harmonics, Hopf algebras, and Polytopes

The starting point for this project is a surprising relationship that we recently discovered between diagonal harmonics, Hopf algebras, and polytope theory. Studying the unexpected connections between these seemingly disparate fields requires a broad range of expertise across different areas, including algebraic combinatorics, discrete geometry, algebra, representation theory, and symmetric functions. We propose to forge novel connections between these areas by attacking a selection of open problems relating them. Our trilateral approach will open new, unexplored avenues for future research. Our analysis will require the implementation of algorithms that will contribute to the development of free open source software.